====== Implicit Competition ======

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Here we extend the analysis in [[..:continuous_types]] to support implicit((To be defined.)) (as well as explicit) competition.

===== Logistic processes =====

Logistic dynamics arise from an asexual birth process (like \(N \rightarrow 2 N\)) and an //explicit competition// process: \(2 N \rightarrow N\).  But another way to arrive at the logistic equation is through //implicit competition// where individuals don't die due to one-on-one interactions directly, but because they don't have a resource (eg. space) necessary for reproduction.

To capture this process we need to track the resource availability, \(V\).  Following the analysis in [[..:continuous_types]]

\[
\begin{array}{rcll}
  N + V & \xrightarrow{\alpha} & 2 N     & \text{(implicit competition / asexual reproduction)}  \\
  2 N & \xrightarrow{\chi}   & N + V     & \text{(explicit competition)} \\
  N   & \xrightarrow{\delta} & V         & \text{(spontaneous death)} \\
  N + V & \xrightarrow{\lambda} & 2 V    & \text{(loneliness)}  \\
\end{array}
\]
so
\[
  \diff{n}{t} = (\alpha - \lambda) n v - \chi n^2 - \delta n = - \diff{v}{t}.
\]
Note that the //sexual reproduction// process (\(2 N + V \rightarrow 3 N\)) is deliberately excluded because it adds a cubic term to the rate equation so can't be represented by the logistic equation.  It could work if we make the assumption that the cubic term is negligible but the math gets messy.  It's easier to leave it out.

Since \(n+v=w\) is a conserved quantity we can write
\[
\begin{array}{rl}
  \diff{n}{t} & = (\alpha - \lambda) n (w-n) - \chi n^2 - \delta n \\
              & = ((\alpha - \lambda) w - \delta) n - (\alpha + \chi - \lambda) n^2.
\end{array}
\]
which is just the logistic equation
\[
  \diff{n}{t} = r n (1 - n/K)
\]
with \(r=(\alpha - \lambda) w -\delta\) and \(K=r/(\alpha + \chi - \lambda)\).  We require \(r,K>0\) -- equivalently, \(\alpha w > \lambda w + \delta\) -- for a biologically meaningful outcome, with a stable, positive equilibrium population density.

Note that the above reactions capture //all// possible processes involving a single type of individual, a single limiting resource, and satisfying the following assumptions:
  * individual plus resource density is a conserved quantity -- each individual sequesters a unit of available resource while alive and releases it on death;
  * there is no spontaneous creation / immigration -- an individual \(N\) is a required reactant in any birth process;
  * no higher order than pairwise interactions; and
  * only one individual changes per reaction -- eg. one birth or death.

===== Extension to two types =====

Let's consider the extension of the above to two different types, \(A\) and \(B\).  We make the additional assumptions:
  * there are two types of individuals, \(A\) and \(B\); and
  * only the focal reactant is identified by type, any partner type is neglected.  (Equivalently, the type of the partner is irrelevant to the reaction rate constant, as explained in [[#Appendix B: Focal individual]].)

Then the four reactions above expand to the following 14 reactions:
\[
\begin{array}{rcll}
  A + V  & \xrightarrow{\alpha_{aa}} & A + A       & \text{(implicit competition / asexual reproduction)}  \\
  A + V  & \xrightarrow{\alpha_{ab}} & A + B       & \\
  B + V  & \xrightarrow{\alpha_{ba}} & B + A       & \\
  B + V  & \xrightarrow{\alpha_{bb}} & B + B       & \\
  \\

  A + N & \xrightarrow{\chi_a}   & N + V        & \text{(explicit competition)} \\
  B + N & \xrightarrow{\chi_b}   & N + V         & \\
  \\

  A   & \xrightarrow{\delta_a} & V       & \text{(spontaneous death)}  \\
  B   & \xrightarrow{\delta_b} & V      & \\
  \\
  
  A + V  & \xrightarrow{\lambda_a} & 2 V       & \text{(isolation)}  \\
  B + V  & \xrightarrow{\lambda_b} & 2 V       & \\
  \\

  A   & \xrightarrow{\phi_a} & B             & \text{(spontaneous transition)}  \\
  B   & \xrightarrow{\phi_b} & A             & \\
  \\

  A + N & \xrightarrow{\nu_a}   & B + N         & \text{(induced transition)} \\
  B + N & \xrightarrow{\nu_b}   & A + N         &
\end{array}
\]

The first 10 reactions are extensions of the logistic processes.  The last four are new, representing transitions between the two types((These reactions may be considered implicit in the logistic processes.)).

===== Dynamics =====

Let's look at the dynamics of the two types.  From the [[wp>law of mass action]] their densities, \(a\) and \(b\), follow
\[
\begin{array}{rcl}
  \diff{a}{t} & = & \alpha_{aa} a v + \alpha_{ba} b v - \chi_a a n - \delta_a a - \lambda_a a v - \phi_a a + \phi_b b - \nu_a a n + \nu_b b n \\
  \diff{b}{t} & = & \alpha_{ab} a v + \alpha_{bb} b v - \chi_b b n - \delta_b b - \lambda_b b v + \phi_a a - \phi_b b + \nu_a a n - \nu_b b n
\end{array}
\]
where \(n=a+b\) and \(w=n+v\).

We're really interested in the dynamics of the total population density so let's switch to describing the density \(n\) and the frequency of \(a\)'s, \(f=a/n\):
\[
\begin{array}{rcl}
  \diff{n}{t} & = & \diff{a}{t} + \diff{b}{t} \\
  \diff{f}{t} & = & \frac{1}{n} \diff{a}{t} - \frac{f}{n} \diff{n}{t}.
\end{array}
\]
With substitution and some algebra we find
\[
\begin{array}{rl}
  \diff{n}{t} = & n \left[ w \left( f (\alpha_{aa} + \alpha_{ab} - \lambda_a) + (1-f) (\alpha_{ba} + \alpha_{bb} - \lambda_b) \right) - f \delta_a - (1-f) \delta_b \right] \\
                & - n^2 \left[ f (\alpha_{aa} + \alpha_{ab} + \chi_a - \lambda_a) + (1-f) (\alpha_{ba} + \alpha_{bb} + \chi_b - \lambda_b) \right] \\
  \diff{f}{t} = & w \left[ - f^2 \alpha_{ab} + f(1-f)(\alpha_{aa} + \lambda_b - \alpha_{bb} - \lambda_a) + (1-f)^2 \alpha_{ba}  \right] \\
                & + f(1-f) (\delta_b - \delta_a) - f \phi_a + (1-f) \phi_b \\
                & + n \left[ 
                  \begin{split}
                    f^2 \alpha_{ab} + f(1-f)(\alpha_{bb} + \lambda_a + \chi_b - \alpha_{aa} - \lambda_b - \chi_a) \\
                     - (1-f)^2 \alpha_{ba} - f \nu_a + (1-f) \nu_b
                  \end{split}
                  \right].
\end{array}
\]

===== Separation of timescales =====

The full dynamics may be difficult to characterize.  But it is more manageable if we assume that the dynamical variables, \(n\) and \(f\), change at drastically different speeds.  For example, we can see immediately, from \(\diff{n}{t} = n(\cdot) - n^2 (\cdot)\) that if \(n\) equilibrates much faster than \(f\) -- so that \(f\) is effectively constant over the short time \(n\) equilibrates -- then the population dynamics reduce to logistic, with a slowly varying carrying capacity determined by \(f\)((I should look into this more carefully: \(n'=\diff{n}{f} f'\).)).

On the other hand, and more formally, let's consider the case where the "ecological" processes((Those affecting the population density, \(n\).)) are very slow:
\[
\begin{array}{rlrl}
  \alpha  & = \epsilon \tilde{\alpha}  &  \delta & = \epsilon \tilde{\delta} \\
  \lambda & = \epsilon \tilde{\lambda}   &    \chi & = \epsilon \tilde{\chi}
\end{array}
\]
where \(\epsilon \ll 1\).

On the slow timescale, \(\tilde{t} = \epsilon t\), we have (where \(x' = \diff{x}{\tilde{t}} = \frac{1}{\epsilon}\diff{x}{t}\))
\[
\begin{array}{rl}
  n' = & n \left[ w \left( f(\tilde{\alpha}_{aa} + \tilde{\alpha}_{ab}) + (1-f) (\tilde{\alpha}_{ba} + \tilde{\alpha}_{bb}) \right) - f \tilde{\delta}_a - (1-f) \tilde{\delta}_b \right] \\
       & - n^2 \left[ f \tilde{\chi}_a + (1-f) \tilde{\chi}_b - f (\tilde{\beta}_{aa} + \tilde{\beta}_{ab}) - (1-f) (\tilde{\beta}_{ba} + \tilde{\beta}_{bb}) \right] \\
  f' = & \frac{1}{\epsilon} \left[ - f \phi_a + (1-f) \phi_b - n f \nu_a + n (1-f) \nu_b \right] + O[\epsilon^0].
\end{array}
\]

To lowest order((We can explore higher order corrections later.)) we must then have 
\[
  0 = - f \phi_a + (1-f) \phi_b - n f \nu_a + n (1-f) \nu_b 
\]
so that \(f'\) doesn't diverge.  In this case we find \(f\) quickly equilibrates (zeroth-order QSSA) to
\[
\begin{array}{rcl}
  f^*     & = & \frac{ n \nu_b + \phi_b}{n (\nu_a + \nu_b) + \phi_a + \phi_b} \\
  1 - f^* & = & \frac{ n \nu_a + \phi_a}{n (\nu_a + \nu_b) + \phi_a + \phi_b}.
\end{array}
\]
Note that we don't require //all// of the "physiological" processes (involving \(\nu\) and \(\phi\)) to be fast.  But if they are slow (on the same timescale as \(\alpha,\delta,\chi,\lambda\)) then they are effectively zero in \(f^*\) for the purpose of this zeroth-order approximation.

To show deviations from logistic behaviour we need \(f^*\) to be a non-constant function of \(n\).  Otherwise, the slow dynamics reduce to \(n' = n(\cdot) - n^2(\cdot)\), the logistic equation.  \(f^*\) is constant under the following four conditions: \(\nu_a=\nu_b=0\), \(\nu_a=\phi_a=0\), \(\nu_b=\phi_b=0\), or \(\phi_a=\phi_b=0\).  As long as these pairs of rate constants are not both "slow" we will see dynamics beyond logistic.  Any one of these rate constants can be "slow" or one of the pairs \((\nu_a,\phi_b)\) or \((\nu_b,\phi_a)\) can be slow, but no more, for non-logistic behaviour.

The general form of the slow dynamics are
\[
  n' = \frac{n}{d_0 + d_1 n} \left( u_0 + u_1 n - u_2 n^2 \right)
\]
where((\(d_0, d_1 = 0\) are neglected because they lead to the degenerate logistic dynamics.))
\[
\begin{array}{rl}
  d_0 = & \phi_a + \phi_b > 0 \\
  d_1 = & \nu_a + \nu_b > 0 \\
  u_0 = & ( A_a w - \tilde{\delta}_a ) \phi_b + ( A_b w - \tilde{\delta}_b ) \phi_a \\
  u_1 = & A_a (\nu_b w - \phi_b) + A_b (\nu_a w - \phi_a) \\
        & - \tilde{\delta}_a \nu_b - \tilde{\delta}_b \nu_a - \tilde{\chi}_a \phi_b - \tilde{\chi}_b \phi_a \\
  u_2 = & ( A_a + \tilde{\chi}_a ) \nu_b + ( A_b + \tilde{\chi}_b ) \nu_a
\end{array}
\]
and
\[
\begin{array}{rl}
  A_a & \equiv \tilde{\alpha}_{aa} + \tilde{\alpha}_{ab} - \tilde{\lambda}_a \\
  A_b & \equiv \tilde{\alpha}_{ba} + \tilde{\alpha}_{bb} - \tilde{\lambda}_b.
\end{array}
\]
===== General Allee =====

A common form of the Allee effect is
\[
  n' = r \, n \, a(n) \left( 1 - \frac{n}{K} \right)
\]
where
\[
\begin{array}{rl}
  a(n) & = 1-A + A \frac{n}{n+n_{1/2}} = \frac{n-n_A}{n+n_{1/2}} \\
  n_A  & \equiv (A-1) n_{1/2}.
\end{array}
\]

Without explicitly solving the dynamics we can write our equation for \(n'\) in precisely this form by suitable choice of the parameters \(r, K, A\) and \(n_{1/2}\).  That's handy because these parameters have intuitive meaning.  \(r\) is the intrinsic growth rate (at low density, neglecting the Allee effect); \(K\) is the carrying capacity; \(A\) is the strength of the Allee effect (\(A=0\) means no effect, \(A\leq 1\) is weak, \(A>1\) is strong); \(n_{1/2}\) is the characteristic population density below which the Allee effect is significant.  See [[..:..:general_allee:start]] for more information.

The general Allee form can be recast as
\[
  n' = r \frac{n}{n+n_{1/2}} \left[ n_{1/2} (1-A) + n \left(1 - (1-A) \frac{n_{1/2}}{K}\right) - \frac{n^2}{K} \right].
\]

The mapping of parameters is then
\[
\begin{array}{rl}
  n_{1/2} = & \frac{d_0}{d_1} \\
  K = & \frac{u_1 \mp \sqrt{u_1^2 + 4 u_0 u_2}}{2 u_2} \\
  r = & \frac{u_1 \mp \sqrt{u_1^2 + 4 u_0 u_2}}{2 d_1} \\
  A = & \frac{2 d_0 u_2 + d_1 (u_1 \pm \sqrt{u_1^2 + 4 u_0 u_2})}{2 d_0 u_2}.
\end{array}
\]

==== Alternate formulation ====

The general Allee form is perhaps better cast as
\[
  n' = r \frac{n}{1+n/n_{1/2}} \left(1 - \frac{n}{n_A}\right) \left(1 - \frac{n}{K}\right).
\]
In this formulation the connection with the logistic equation is more direct: when \(n_{1/2},\abs{n_A}\gg \abs{K}\) the dynamics are approximately logistic for \(n \lesssim \abs{K}\).

The two non-trivial equilibria (at \(K\) and \(n_A\)) are given by the roots
\[
  n_\pm \equiv n|_{n'=0} = \frac{u_1 \pm R}{2 u_2}
\]
where
\[
  R \equiv \sqrt{u_1^2 + 4 u_0 u_2}.
\]

But which root should be assigned to each equilibrium?  Looking at all 8 possible combinations where \((u_0, u_1, u_2)\) can be positive or negative we find that in only four cases there exists a positive, stable equilibrium -- given by the positive radical, \(n_+>0\).  Therefore it is natural to identify the carrying capacity with this equilibrium:
\[
  K \equiv n_+ = \frac{u_1 + R}{2 u_2}
\]
and
\[
\begin{array}{ll}
  n_A \equiv n_- & = \frac{u_1 - R}{ 2 u_2} = \frac{u_1}{u_2} - K \\
  n_{1/2} & \equiv \frac{d_0}{d_1} \\
  r & \equiv \frac{u_0}{d_0}.
\end{array}
\]

The sustainable cases occur when a positive equilibrium exists, \(K>0\).  It so happens that in these cases \(K\) is always stable.

^ \(u_0\)  ^  \(u_1\)  ^  \(u_2\)  ^  \(u_1 - R\)  ^  \(u_1 + R\) ^  \(r\)  ^  \(n_A\)  ^  \(K\)  ^                      Stability                       ^  Sustainable  ^    Allee    ^
|    -     |     -     |     -     |       -       |      +       |    -    |     +     |    -    |  \((K \rightarrow)\; 0 \leftarrow n_A \rightarrow\)  |      no       |             |
|    -     |     -     |     +     |       -       |      -       |    -    |     -     |    -    |  \((n_A \leftarrow K \rightarrow)\; 0 \leftarrow\)   |      no       |             |
|    -     |     +     |     -     |       -       |      +       |    -    |     +     |    -    |  \((K \rightarrow)\; 0 \leftarrow n_A \rightarrow\)  |      no       |             |
|    -     |     +     |     +     |       +       |      +       |    -    |     +     |    +    |    \(0 \leftarrow n_A \rightarrow K \leftarrow\)     |      yes      |   strong    |
|    +     |     -     |     -     |       -       |      -       |    +    |     +     |    +    |    \(0 \rightarrow K \leftarrow n_A \rightarrow\)    |      yes      |  "contra"?  |
|    +     |     -     |     +     |       -       |      +       |    +    |     -     |    +    |  \((n_A \leftarrow)\; 0 \rightarrow K \leftarrow\)   |      yes      |    weak?    |
|    +     |     +     |     -     |       +       |      +       |    +    |     -     |    -    |  \((K \rightarrow n_A \leftarrow)\; 0 \rightarrow\)  |      no       |             |
|    +     |     +     |     +     |       -       |      +       |    +    |     -     |    +    |  \((n_A \leftarrow)\; 0 \rightarrow K \leftarrow\)   |      yes      |    weak?    |





===== Nondimensionalization =====

We can simplify the above dynamics by recognizing that we're not interested in the units of the population density or timescale.  We can rescale to units where((Assuming \(r,K>0\).))
\[
\begin{array}{rl}
  x & \equiv n / K \\
  \tau & \equiv r \tilde{t}
\end{array}
\]
so that, for example, \(x=1\) is the equilibrium density (at carrying capacity).  In these units the dynamics become
\[
  \diff{x}{\tau} = \frac{x - x_A}{x+x_{1/2}} x (1-x)
\]
where
\[
\begin{array}{rl}
  x_{1/2} & \equiv n_{1/2} / K \\
  x_A & \equiv (A-1) x_{1/2}.
\end{array}
\]

<div right round 500px box>
<BOOKMARK:figure_1>
{{svg>figure1.svg}}

Figure 1: Parameter space of nondimensionalized implicit competition model.  There is a strong Allee effect when \(x_{1/2},x_A>0\).  The population growth rate diverges at \(x=-x_{1/2}\) which is positive when \(x_{1/2}<0\) so this region of parameter space is assumed to be non-biological.
</div>

Now we see that the behaviour is characterized by two parameters, \(x_{1/2}\) and \(x_A\).  We should exclude from consideration the region \(x_{1/2}<0\) because in that region \(\diff{x}{\tau}\) diverges for \(x>0\), presumably a non-biological dynamic.  When \(x_{1/2}>0\) it just scales the rate by a positive factor without changing the equilibria or stability of the dynamics.

The only bifurcation parameter is \(A\) or, equivalently, \(x_A\).  When \(0<A<1\) the population exhibits a //weak Allee effect// where the growth rate in the limit of small populations is slowed.  On the other hand, when \(A>1\) we find a //strong// Allee effect -- populations below the //Allee density//, \(x<x_A\), are unsustainable.  (When \(A<0\) there is a //contra-Allee// effect that boosts growth rates of small populations.


===== Appendix A: Reverse QSSA =====

In [[#Separation of timescales]] we assumed "ecological" processes were much slower than "physiological".  For completeness, let's reverse that assumption and assume "ecology" is fast:
\[
\begin{array}{rl}
  \phi  & = \epsilon \tilde{\phi} \\
  \nu   & = \epsilon \tilde{\nu}
\end{array}
\]
where \(\epsilon \ll 1\).

On the slow timescale, \(\tilde{t} = \epsilon t\), we have (where \(\diff{x}{t} = \epsilon x'\))
\[
\begin{array}{rl}
  n' = & \frac{n}{\epsilon} \left[
           \begin{split}
             (w-n) \left( f(\alpha_{aa}+\alpha_{ab}-\lambda_{a}) + (1-f)(\alpha_{ba}+\alpha_{bb}-\lambda_{b}) \right) \\
             - f (n \chi_a + \delta_a) - (1-f) (n \chi_b + \delta_b)
           \end{split}  
         \right] \\
  f' = & - f (n \tilde{\nu}_a + \tilde{\phi}_a) + (1-f) (n \tilde{\nu}_b + \tilde{\phi}_b) \\
       & + \frac{1}{\epsilon} \left[
           \begin{split}
             (w-n) \left( f (1-f) \alpha_{aa} - f^2 \alpha_{ab} + (1-f)^2 \alpha_{ba} - f (1-f) \alpha_{bb} \right) \\
             + f (1-f) (\delta_b - \delta_a) + f (1-f) (w-n) (\lambda_b - \lambda_a)\\
             + f (1-f) n (\chi_b - \chi_a)
           \end{split}
         \right].  
\end{array}
\]

Notice both expressions contain an \(\epsilon^{-1}\) term.  To prevent divergence as \(\epsilon\rightarrow 0\) we need to satisfy two conditions, which in general is only true if both \(n\) and \(f\) are at quasi-steady states.  This won't give us what we need: a slowly-varying dynamical variable.

To solve it, I think we need to transform our original densities, \(a\) and \(b\), into two new variables where one of them depends only on the "slow parameters", \(\phi\) and \(\nu\).  How can we find that?  --- //[[rik.blok@ubc.ca|Rik Blok]] 2011-07-08 14:45//

It's an ugly hack but here's a technique that will give the slow dynamics for \(n\) if we assume that "ecology" is fast:
  - If "ecology" is fast then \(n\) equilibrates quickly so we get a relationship between \(f\) and \(n\) by solving \(\diff{n}{t}=0\).  Find \(f=f^*(n)\) in equilibrium.  (Although it doesn't look like it we're setting \(n\) here.)
  - In equilibrium \(n\) doesn't depend explicitly on time anymore, but only implicitly through \(f\).  So \(\diff{n}{t} = \diff{n}{f} \diff{f}{t}\).
  - Turn things around so we're replacing \(f\)'s with \(n\)'s: \[ \diff{n}{t} = \frac{1}{\diff{f^*}{n}} \left. \diff{f}{t} \right|_{f=f^*} \]
  - If I did the calculation right you should find \[ \diff{n}{t} = (c_0 + c_1 n) ( d_0 + d_1 n + d_2 n^2). \]

 --- //[[rik.blok@ubc.ca|Rik Blok]] 2011-07-15 20:55//

===== Appendix B: Focal individual =====

Note that the assumption of having a single "focal individual" and disregarding the "type" of its partner isn't necessary. We arrive at the same result if we just assume that the rate constants are identical, regardless of the partner type.  Consider, for example, the explicit competition process
\[
   A + N \xrightarrow{\chi_a} N + V
\]
which contributes the following term to the rate equation for \(A\)-types:
\[
  \diff{a}{t} = \cdots - \chi_a a n.
\]
But since \(n = a + b\) this is the same contribution as would be made by the following pair of processes:
\[
\begin{array}{rcl}
  A + A & \xrightarrow{\chi_{aa}}   & A + V \\
  A + B & \xrightarrow{\chi_{ab}}   & B + V
\end{array}
\]
if \(\chi_{aa} = \chi_{ab} \equiv \chi_a\).  So we don't need a "focal individual", it is sufficient if the reaction rate constants are such that the type of the partner is irrelevant.